Train - From Aberdeen it is the first stop South after Portlethen, and costs less than 5 pounds round trip.
Bus 45-60 minute trip from Aberdeen even less expensive than the train. Get the 107,x7 or 117 from the Aberdeen bus station, next to the train station
Stonehaven is a very small town, so walking is your best bet
There are taxis in the area
Dunnottar Castle, . By far one of the most amazing sites in Scotland. The castle is perched on cliffs overhanging the North Sea and provides stunning views. The best way to get there for adventurous folk is the mile-long hike from the town of Stonehaven. The hike is a narrow path along the cliffs that provides stunning views the entire way. Definitely a must-see.
Stonehaven Harbour - The harbour forms what I would call the town center. There is a collection of local pubs and restaurants, though as it is a small town, we are talking about maybe 4 or 5 places. However, having a pint at a table overlooking the harbor and hanging with locals is a great experience. This town is very small, and in the town there are very few travelers, so it is a great chance to meet and talk to locals.
Play golf at the Stonehaven Golf Course.  Probably one of the most tremendous golf courses you will play in your lifetime. Absolutely stunning.
Run and Swim with Splash'n'Dash from the Leisure Centre on Tues/Thurs evenings 
There are a couple of gift shops actually - Purdies, in the Market Square sells an eclectic mix of gifts, cards, jewellery and shoes.
Charles Michie, also in the Market Square is a pharmacy, but sells trinkets, gifts, ornaments etc.
The Shell Hoosie at the harbour sells sea/marine based gifts as well as ice creams, coffee to take away etc.
Mimi's, in the Market Square does the best hot chocolate around!
Here and Now, in Barclay Street, has lovely homebakes, sandiwches and snacks. Also sells fairtrade goods, gifts etc.
Gino's, just up from the Market Square is a wee Italian run cafe, and sells good quality food at reasonable prices.
Molly Gunnings, at the beach is open all day and sells good Scottish breakfasts, as well as the usual homebakes, paninis etc.
The Tollbooth, in the harbour, . For World Class seafood.